1. Anomalous excitation spectra of frustrated quantum antiferromagnets John Fjaerestad University of Queensland Work done in collaboration with: Weihong Zheng, UNSW Rajiv Singh, UC Davis Ross McKenzie, UQ Radu Coldea, Oxford (Bristol) Reference: cond-mat/0506400 (Phys. Rev. Lett., in press)

2. Outline Two-dimensional antiferromagnets In systems with a magnetically ordered ground state, semiclassical spin-wave theory has been believed to give a good account of the ground state and excitations Will show that in frustrated systems with magnetically ordered ground state, the high-energy excitations can show strong deviations from spin-wave theory Will suggest a possible interpretation of the excitation spectrum of the triangular antiferromagnet

3. Important “minimal” model: Heisenberg model Two-dimensional antiferromagnets Here we will focus on S=1/2

4. 2D S=1/2 Heisenberg antiferromagnet on square lattice Low-energy effective model for the cuprate superconductors (stacked layers of 2D CuO2 planes) at zero doping Ground state: Long-range magnetic (Neel)order Excitations: Dispersion for S=1 magnons Circles: Data points for Cu(DCOO) 2 4D 2 0 (Christensen et al (2004)) Full line: Linear spin-wave theory Dashed line: Series expansion result (Singh & Gelfand (1995))

5. Spin-wave theory: semiclassical approach to magnetically ordered systems Spin waves: weak oscillations of the spins around their classical direction Classical spins: Consider quantity A(S) (energy, magnetization) i.e. effects of quantum fluctuations enter as correction terms to the classical result in powers of 1/S S=1/2 square lattice model: Ground state and excitations well described by spin-wave theory (except for small deviations in high-energy spectra)

6. ? Introduce frustration Quantum fluctuations, enhanced by frustration, may lead to a magnetically disordered ground state (spin liquid) (Anderson 1973, Fazekas and Anderson 1974) A less exotic, but still very interesting possibility: Ground state is magnetically ordered, but magnon dispersion shows large deviations from spin-wave theory What can be done to increase the impact of quantum fluctuations? The fundamental excitations are spinons with S=1/2 (“fractionalized” compared to the S=1 of conventional magnons)

7. S = ½ Heisenberg antiferromagnet on an anisotropic triangular lattice J 1 /J 2 = 0 : Square lattice model (old results) J 1 /J 2 < 0.7 J 1 /J 2 = 1 : Triangular lattice model J 1 /J 2 = 3 : Cs 2 CuCl 4 Focus here:

8. Frustrated Neel phase (0 < J 1 /J 2 < 0.7) Magnon dispersion for different values of J 1 /J 2 (Zheng et al., cond-mat/0506400) (p/2,p/2) As J 1 /J 2 is increased, the local minimum at  becomes more pronounced and the energy difference between  and  increases In contrast, linear spin-wave theory predicts no energy difference between  and  (Merino et al., 1999)

9. A B C O A Q D Triangular lattice model (J 1 = J 2 ) kyky kxkx Magnon dispersion LSWT Series Large deviations from linear spin-wave theory (LSWT) at high energies Ground state: Magnetically ordered (120 ° pattern)

10. Cs 2 CuCl 4 (J 1 = 3J 2 ) (Dominant continuum scattering indicative of spin liquid physics at high energies (Coldea et al, PRB 2003)) With respect to the linear spin-wave theory (LSWT) curve, the dispersion is strongly enhanced along the strong (J 1 ) bonds and decreased perpendicular to them Quantum fluctuations make the dispersion look more one-dimensional Series LSWT Exp Nonlinear spin-wave theory is also not able to account quantitatively for this large quantum renormalization (Veillette et al., PRB 72, 134429 (2005)

11. In frustrated systems with magnetically ordered ground state, high-energy excitations can show strong deviations from spin-wave theory Main message so far: In remaining part of the talk: Present a possible interpretation of the excitation spectra for the triangular antiferromagnet (J 1 =J 2 )

12. Square lattice model (J 1 = 0) Excitations: Dispersion for S=1 magnons Circles: Data points for Cu(DCOO) 2 4D 2 0 (Christensen et al, JMMM 272-276, 896 (2004)) Full line: Linear spin-wave theory Dashed line: Series expansion result (Singh & Gelfand, Phys. Rev. B 52, 15695 (1995)) Local minimum in dispersion at ( ,0) : not captured by spin-wave theory Interpreted as signature of RVB physics (Hsu, Phys. Rev. B 41, 11379 (1990); Syljuasen and Ronnow, J. Phys. Condens. Matter 12, L405 (2000), Ho et al, Phys. Rev. Lett. 86, 1626 (2001))

13. Description of resonating-valence-bond (RVB) states (Anderson) The S=1/2 Heisenberg model is the U/t  limit of the fermionic Hubbard model at half-filling Ground state of a mean-field (i.e. quadratic) fermionic Hamiltonian RVB state Gutzwiller projection: Projects onto space of singly-occupied sites  -flux phase: Ground state of mean-field Hamiltonian describing fermions hopping on a square lattice with fictitious flux ±  threading alternating plaquettes  -flux phase spinon dispersion: Gapless excitations at      

14. “  -flux state” RVB solution for square lattice model (J 1 =0) (Affleck and Marston 1988, Kotliar 1988) Projected  -flux state: E = -0.319 J/bond Incorporate magnetic long-range order into  -flux state (Hsu, PRB 1990): E = -0.332 J/bond S=1 magnon dispersion (Hsu, PRB 1990): Magnon is bound state of two spinons Ground state energies: True ground state (Neel-ordered) E = -0.3346 J/bond Deep minimum at ( ,0)  points where magnon dispersion has local minima  points where spinon dispersion of  -flux phase has gapless excitations Two-spinon (particle-hole) continuum also has minima at  points: pushes down magnon energy

15. A B C O A Q D Triangular lattice model (J 1 = J 2 ) kyky kxkx Magnon dispersion LSWT Series Large deviations from linear spin-wave theory (LSWT) at high energies The two-spinon continuum has local minima at the k- vectors showing the largest deviations from LSWT Assumed locations of minima of spinon dispersion

16. Conclusions Deviations from LSWT at high energies increase with frustration in Neel phase Very strong deviations from LSWT in triangular-lattice model leading to local minima and flat regions in dispersion Interpreted in terms of two-spinon picture Quantum fluctuations make Cs 2 CuCl 4 dispersion look more one-dimensional

17. Strongly correlated electrons Electron-electron interactions very strong Can lead to qualitatively different behavior from systems of noninteracting or weakly interacting electrons What kinds of unconventional physics can arise as a consequence of the strong correlations? Unconventional ground states and/or Unconventional excitations

18. The Hubbard model (“minimal” model for correlated electrons on a lattice) H = ¡ t P h i ; j i ; ¾ ( c y i ; ¾ c j ; ¾ + h. c. ) + U P i n i ; " n i ; # Kinetic energyInteraction energy Fundamental competition between the delocalization favored by the hopping term (K.E.) and the tendency of the on-site repulsion term (I.E.) to localize electrons on sites When U/t    and # of electrons = # of sites (half-filling) Low-energy states have no doubly occupied sites (i.e. no charge fluctuations) The system is a (Mott) insulator